Add menu entry to generic right-click menu?
As of this commit, thanks to @p2or and @brecht, the bpy.ops.ui.copy_data_path_button() operator now allows to copy the full data-path of the active property under the mouse cursor to the clipboard  which can be used to assign custom values to the active property (only available in latest versions).



Simplified version of the provided example in the patch (integers only):

import bpy

class MouseCursorPropertyOperator(bpy.types.Operator):
    bl_idname = "screen.active_int_property_add"
    bl_label = "Adjust active mouse cursor integer property"
    bl_options = {'REGISTER', 'UNDO'}

    my_int = bpy.props.IntProperty(default=100)

    @classmethod
    def poll(cls, context):
        return bpy.ops.ui.copy_data_path_button.poll()

    def execute(self, context):      

        # get the data path
        bpy.ops.ui.copy_data_path_button()
        path = context.window_manager.clipboard

        # get full data path
        bpy.ops.ui.copy_data_path_button(full_path=True)
        full_path = context.window_manager.clipboard

        # split path in class and property
        rna, prop = context.window_manager.clipboard.rsplit('.', 1)

        # set attribute if type is integer
        if type(eval(full_path)) is int:
            rna_eval = (eval(rna))
            value = getattr(rna_eval, prop)
            setattr(rna_eval, prop, value + self.my_int)
            self.report({"INFO"}, "{} set to {}".format(prop, value + self.my_int))
        else:
            print (type(eval(full_path)))
            self.report({"INFO"}, "{} is not an integer property".format(prop))

        return {'FINISHED'}


addon_keymaps = []

def register():
    bpy.utils.register_class(MouseCursorPropertyOperator)

    # handle the keymap
    wm = bpy.context.window_manager
    kc = wm.keyconfigs.addon
    if kc:
        km = wm.keyconfigs.addon.keymaps.new(name='Screen')
        kmi = km.keymap_items.new(MouseCursorPropertyOperator.bl_idname, type='Q', value='PRESS', shift=True)
        addon_keymaps.append((km, kmi))

def unregister():

    for km, kmi in addon_keymaps:
        km.keymap_items.remove(kmi)
    addon_keymaps.clear()

    bpy.utils.unregister_class(MouseCursorPropertyOperator)


if __name__ == "__main__":
    register()




And as of this commit, thanks to @plasmasolutions you can now add an operator to the menu:



Provided example of the patch: 

import bpy
from bpy.types import Header, Menu, Panel

def dump(obj, text):
    print('-'*40, text, '-'*40)
    for attr in dir(obj):
        if hasattr( obj, attr ):
            print( "obj.%s = %s" % (attr, getattr(obj, attr)))

class TestOp(bpy.types.Operator):
    """Tooltip"""
    bl_idname = "object.test_op"
    bl_label = "Execute a custom action"

    @classmethod
    def poll(cls, context):
        return context.active_object is not None

    def execute(self, context):
        if hasattr(context, 'button_pointer'):
            btn = context.button_pointer 
            dump(btn, 'button_pointer')

        if hasattr(context, 'button_prop'):
            prop = context.button_prop
            dump(prop, 'button_prop')

        if hasattr(context, 'button_operator'):
            op = context.button_operator
            dump(op, 'button_operator')     

        return {'FINISHED'}

# This class has to be exactly named like that to insert an entry in the right click menu
class WM_MT_button_context(Menu):
    bl_label = "Add Viddyoze Tag"

    def draw(self, context):
        pass

def menu_func(self, context):
    layout = self.layout
    layout.separator()
    layout.operator(TestOp.bl_idname)

def register():
    bpy.types.WM_MT_button_context.append(menu_func)

def unregister():
    bpy.types.WM_MT_button_context.remove(menu_func)

classes = (
    TestOp,
    WM_MT_button_context,
)

if __name__ == "__main__":
    from bpy.utils import register_class
    for cls in classes:
        register_class(cls)
    register()